To find the number of solutions for the equation \( 12(x - 3) + 18 = \text{RHS} \), we first simplify the left-hand side (LHS):
\[ 12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18 \]
So the equation becomes:
\[ 12x - 18 = \text{RHS} \]
Now, we will evaluate each expression on the right-hand side to see how many solutions each would yield.
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For \( \text{RHS} = 6(2x - 3) \): \[ 6(2x - 3) = 12x - 18 \] Setting the LHS equal to RHS: \[ 12x - 18 = 12x - 18 \] This is an identity, meaning it holds true for all \( x \). Therefore, we have C) Infinite Solutions.
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For \( \text{RHS} = 4(3x - 3) \): \[ 4(3x - 3) = 12x - 12 \] Setting the LHS equal to RHS: \[ 12x - 18 = 12x - 12 \] Rearranging gives: \[ -18 \neq -12 \] This is a contradiction, implying there are B) No Solution.
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For \( \text{RHS} = 4(2x - 3) \): \[ 4(2x - 3) = 8x - 12 \] Setting the LHS equal to RHS: \[ 12x - 18 = 8x - 12 \] Rearranging gives: \[ 12x - 8x = -12 + 18 \implies 4x = 6 \implies x = \frac{3}{2} \] Therefore, we have A) One Solution.
In summary:
- \( 6(2x - 3) \): C) Infinite Solutions
- \( 4(3x - 3) \): B) No Solution
- \( 4(2x - 3) \): A) One Solution
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